Jmat.Real.erfi, branch x less than or equal to 0.5

Percentage Accurate: 99.8% → 99.9%

Time: 10.1s

Alternatives: 8

Speedup: 3.1×

• 28.5% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

\[x \leq 0.5\]

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\\ t_1 := \left(t_0 \cdot \left|x\right|\right) \cdot \left|x\right|\\ \left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \frac{2}{3} \cdot t_0\right) + \frac{1}{5} \cdot t_1\right) + \frac{1}{21} \cdot \left(\left(t_1 \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right| \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (* (* (fabs x) (fabs x)) (fabs x)))
        (t_1 (* (* t_0 (fabs x)) (fabs x))))
   (fabs
    (*
     (/ 1.0 (sqrt PI))
     (+
      (+ (+ (* 2.0 (fabs x)) (* (/ 2.0 3.0) t_0)) (* (/ 1.0 5.0) t_1))
      (* (/ 1.0 21.0) (* (* t_1 (fabs x)) (fabs x))))))))

C

double code(double x) {
	double t_0 = (fabs(x) * fabs(x)) * fabs(x);
	double t_1 = (t_0 * fabs(x)) * fabs(x);
	return fabs(((1.0 / sqrt(((double) M_PI))) * ((((2.0 * fabs(x)) + ((2.0 / 3.0) * t_0)) + ((1.0 / 5.0) * t_1)) + ((1.0 / 21.0) * ((t_1 * fabs(x)) * fabs(x))))));
}

Java

public static double code(double x) {
	double t_0 = (Math.abs(x) * Math.abs(x)) * Math.abs(x);
	double t_1 = (t_0 * Math.abs(x)) * Math.abs(x);
	return Math.abs(((1.0 / Math.sqrt(Math.PI)) * ((((2.0 * Math.abs(x)) + ((2.0 / 3.0) * t_0)) + ((1.0 / 5.0) * t_1)) + ((1.0 / 21.0) * ((t_1 * Math.abs(x)) * Math.abs(x))))));
}

Python

def code(x):
	t_0 = (math.fabs(x) * math.fabs(x)) * math.fabs(x)
	t_1 = (t_0 * math.fabs(x)) * math.fabs(x)
	return math.fabs(((1.0 / math.sqrt(math.pi)) * ((((2.0 * math.fabs(x)) + ((2.0 / 3.0) * t_0)) + ((1.0 / 5.0) * t_1)) + ((1.0 / 21.0) * ((t_1 * math.fabs(x)) * math.fabs(x))))))

Julia

function code(x)
	t_0 = Float64(Float64(abs(x) * abs(x)) * abs(x))
	t_1 = Float64(Float64(t_0 * abs(x)) * abs(x))
	return abs(Float64(Float64(1.0 / sqrt(pi)) * Float64(Float64(Float64(Float64(2.0 * abs(x)) + Float64(Float64(2.0 / 3.0) * t_0)) + Float64(Float64(1.0 / 5.0) * t_1)) + Float64(Float64(1.0 / 21.0) * Float64(Float64(t_1 * abs(x)) * abs(x))))))
end

MATLAB

function tmp = code(x)
	t_0 = (abs(x) * abs(x)) * abs(x);
	t_1 = (t_0 * abs(x)) * abs(x);
	tmp = abs(((1.0 / sqrt(pi)) * ((((2.0 * abs(x)) + ((2.0 / 3.0) * t_0)) + ((1.0 / 5.0) * t_1)) + ((1.0 / 21.0) * ((t_1 * abs(x)) * abs(x))))));
end

Wolfram

code[x_] := Block[{t$95$0 = N[(N[(N[Abs[x], $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(t$95$0 * N[Abs[x], $MachinePrecision]), $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision]}, N[Abs[N[(N[(1.0 / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[(2.0 * N[Abs[x], $MachinePrecision]), $MachinePrecision] + N[(N[(2.0 / 3.0), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision] + N[(N[(1.0 / 5.0), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision] + N[(N[(1.0 / 21.0), $MachinePrecision] * N[(N[(t$95$1 * N[Abs[x], $MachinePrecision]), $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Initial Program: 99.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\\ t_1 := \left(t_0 \cdot \left|x\right|\right) \cdot \left|x\right|\\ \left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \frac{2}{3} \cdot t_0\right) + \frac{1}{5} \cdot t_1\right) + \frac{1}{21} \cdot \left(\left(t_1 \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right| \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (* (* (fabs x) (fabs x)) (fabs x)))
        (t_1 (* (* t_0 (fabs x)) (fabs x))))
   (fabs
    (*
     (/ 1.0 (sqrt PI))
     (+
      (+ (+ (* 2.0 (fabs x)) (* (/ 2.0 3.0) t_0)) (* (/ 1.0 5.0) t_1))
      (* (/ 1.0 21.0) (* (* t_1 (fabs x)) (fabs x))))))))

C

double code(double x) {
	double t_0 = (fabs(x) * fabs(x)) * fabs(x);
	double t_1 = (t_0 * fabs(x)) * fabs(x);
	return fabs(((1.0 / sqrt(((double) M_PI))) * ((((2.0 * fabs(x)) + ((2.0 / 3.0) * t_0)) + ((1.0 / 5.0) * t_1)) + ((1.0 / 21.0) * ((t_1 * fabs(x)) * fabs(x))))));
}

Java

public static double code(double x) {
	double t_0 = (Math.abs(x) * Math.abs(x)) * Math.abs(x);
	double t_1 = (t_0 * Math.abs(x)) * Math.abs(x);
	return Math.abs(((1.0 / Math.sqrt(Math.PI)) * ((((2.0 * Math.abs(x)) + ((2.0 / 3.0) * t_0)) + ((1.0 / 5.0) * t_1)) + ((1.0 / 21.0) * ((t_1 * Math.abs(x)) * Math.abs(x))))));
}

Python

def code(x):
	t_0 = (math.fabs(x) * math.fabs(x)) * math.fabs(x)
	t_1 = (t_0 * math.fabs(x)) * math.fabs(x)
	return math.fabs(((1.0 / math.sqrt(math.pi)) * ((((2.0 * math.fabs(x)) + ((2.0 / 3.0) * t_0)) + ((1.0 / 5.0) * t_1)) + ((1.0 / 21.0) * ((t_1 * math.fabs(x)) * math.fabs(x))))))

Julia

function code(x)
	t_0 = Float64(Float64(abs(x) * abs(x)) * abs(x))
	t_1 = Float64(Float64(t_0 * abs(x)) * abs(x))
	return abs(Float64(Float64(1.0 / sqrt(pi)) * Float64(Float64(Float64(Float64(2.0 * abs(x)) + Float64(Float64(2.0 / 3.0) * t_0)) + Float64(Float64(1.0 / 5.0) * t_1)) + Float64(Float64(1.0 / 21.0) * Float64(Float64(t_1 * abs(x)) * abs(x))))))
end

MATLAB

function tmp = code(x)
	t_0 = (abs(x) * abs(x)) * abs(x);
	t_1 = (t_0 * abs(x)) * abs(x);
	tmp = abs(((1.0 / sqrt(pi)) * ((((2.0 * abs(x)) + ((2.0 / 3.0) * t_0)) + ((1.0 / 5.0) * t_1)) + ((1.0 / 21.0) * ((t_1 * abs(x)) * abs(x))))));
end

Wolfram

code[x_] := Block[{t$95$0 = N[(N[(N[Abs[x], $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(t$95$0 * N[Abs[x], $MachinePrecision]), $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision]}, N[Abs[N[(N[(1.0 / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[(2.0 * N[Abs[x], $MachinePrecision]), $MachinePrecision] + N[(N[(2.0 / 3.0), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision] + N[(N[(1.0 / 5.0), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision] + N[(N[(1.0 / 21.0), $MachinePrecision] * N[(N[(t$95$1 * N[Abs[x], $MachinePrecision]), $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Alternative 1: 99.9% accurate, 3.1× speedup

Math

\[\begin{array}{l} \\ \left|\left(x \cdot {\pi}^{-0.5}\right) \cdot \left(\left(0.2 \cdot {x}^{4} + 0.047619047619047616 \cdot {x}^{6}\right) + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)\right)\right| \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (fabs
  (*
   (* x (pow PI -0.5))
   (+
    (+ (* 0.2 (pow x 4.0)) (* 0.047619047619047616 (pow x 6.0)))
    (fma 0.6666666666666666 (* x x) 2.0)))))

C

double code(double x) {
	return fabs(((x * pow(((double) M_PI), -0.5)) * (((0.2 * pow(x, 4.0)) + (0.047619047619047616 * pow(x, 6.0))) + fma(0.6666666666666666, (x * x), 2.0))));
}

Julia

function code(x)
	return abs(Float64(Float64(x * (pi ^ -0.5)) * Float64(Float64(Float64(0.2 * (x ^ 4.0)) + Float64(0.047619047619047616 * (x ^ 6.0))) + fma(0.6666666666666666, Float64(x * x), 2.0))))
end

Wolfram

code[x_] := N[Abs[N[(N[(x * N[Power[Pi, -0.5], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(0.2 * N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision] + N[(0.047619047619047616 * N[Power[x, 6.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.6666666666666666 * N[(x * x), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 99.8%

   \[\left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \frac{2}{3} \cdot \left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{5} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{21} \cdot \left(\left(\left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right| \]

3. Simplified 99.5%

   \[\leadsto \color{blue}{\left|\frac{\left|x\right|}{\sqrt{\pi}} \cdot \left(\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)\right)\right|} \]
4. Step-by-step derivation

   1. div-inv 99.8%

      \[\leadsto \left|\color{blue}{\left(\left|x\right| \cdot \frac{1}{\sqrt{\pi}}\right)} \cdot \left(\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)\right)\right| \]

   2. pow1/2 99.8%

      \[\leadsto \left|\left(\left|x\right| \cdot \frac{1}{\color{blue}{{\pi}^{0.5}}}\right) \cdot \left(\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)\right)\right| \]

   3. pow-flip 99.8%

      \[\leadsto \left|\left(\left|x\right| \cdot \color{blue}{{\pi}^{\left(-0.5\right)}}\right) \cdot \left(\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)\right)\right| \]

   4. metadata-eval 99.8%

      \[\leadsto \left|\left(\left|x\right| \cdot {\pi}^{\color{blue}{-0.5}}\right) \cdot \left(\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)\right)\right| \]

5. Applied egg-rr 99.8%

   \[\leadsto \left|\color{blue}{\left(\left|x\right| \cdot {\pi}^{-0.5}\right)} \cdot \left(\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)\right)\right| \]
6. Step-by-step derivation

   1. unpow1 99.8%

      \[\leadsto \left|\left(\left|\color{blue}{{x}^{1}}\right| \cdot {\pi}^{-0.5}\right) \cdot \left(\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)\right)\right| \]

   2. sqr-pow 30.6%

      \[\leadsto \left|\left(\left|\color{blue}{{x}^{\left(\frac{1}{2}\right)} \cdot {x}^{\left(\frac{1}{2}\right)}}\right| \cdot {\pi}^{-0.5}\right) \cdot \left(\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)\right)\right| \]

   3. fabs-sqr 30.6%

      \[\leadsto \left|\left(\color{blue}{\left({x}^{\left(\frac{1}{2}\right)} \cdot {x}^{\left(\frac{1}{2}\right)}\right)} \cdot {\pi}^{-0.5}\right) \cdot \left(\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)\right)\right| \]

   4. sqr-pow 99.8%

      \[\leadsto \left|\left(\color{blue}{{x}^{1}} \cdot {\pi}^{-0.5}\right) \cdot \left(\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)\right)\right| \]

   5. unpow1 99.8%

      \[\leadsto \left|\left(\color{blue}{x} \cdot {\pi}^{-0.5}\right) \cdot \left(\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)\right)\right| \]

7. Simplified 99.8%

   \[\leadsto \left|\color{blue}{\left(x \cdot {\pi}^{-0.5}\right)} \cdot \left(\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)\right)\right| \]
8. Step-by-step derivation

   1. fma-udef 99.8%

      \[\leadsto \left|\left(x \cdot {\pi}^{-0.5}\right) \cdot \left(\color{blue}{\left(0.2 \cdot {x}^{4} + 0.047619047619047616 \cdot {x}^{6}\right)} + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)\right)\right| \]

9. Applied egg-rr 99.8%

   \[\leadsto \left|\left(x \cdot {\pi}^{-0.5}\right) \cdot \left(\color{blue}{\left(0.2 \cdot {x}^{4} + 0.047619047619047616 \cdot {x}^{6}\right)} + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)\right)\right| \]

10. Final simplification 99.8%

    \[\leadsto \left|\left(x \cdot {\pi}^{-0.5}\right) \cdot \left(\left(0.2 \cdot {x}^{4} + 0.047619047619047616 \cdot {x}^{6}\right) + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)\right)\right| \]

Alternative 2: 99.4% accurate, 3.2× speedup

Math

\[\begin{array}{l} \\ \left|\left(\left(0.2 \cdot {x}^{4} + 0.047619047619047616 \cdot {x}^{6}\right) + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)\right) \cdot \frac{x}{\sqrt{\pi}}\right| \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (fabs
  (*
   (+
    (+ (* 0.2 (pow x 4.0)) (* 0.047619047619047616 (pow x 6.0)))
    (fma 0.6666666666666666 (* x x) 2.0))
   (/ x (sqrt PI)))))

C

double code(double x) {
	return fabs(((((0.2 * pow(x, 4.0)) + (0.047619047619047616 * pow(x, 6.0))) + fma(0.6666666666666666, (x * x), 2.0)) * (x / sqrt(((double) M_PI)))));
}

Julia

function code(x)
	return abs(Float64(Float64(Float64(Float64(0.2 * (x ^ 4.0)) + Float64(0.047619047619047616 * (x ^ 6.0))) + fma(0.6666666666666666, Float64(x * x), 2.0)) * Float64(x / sqrt(pi))))
end

Wolfram

code[x_] := N[Abs[N[(N[(N[(N[(0.2 * N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision] + N[(0.047619047619047616 * N[Power[x, 6.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.6666666666666666 * N[(x * x), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision] * N[(x / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 99.8%

   \[\left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \frac{2}{3} \cdot \left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{5} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{21} \cdot \left(\left(\left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right| \]

3. Simplified 99.5%

   \[\leadsto \color{blue}{\left|\frac{\left|x\right|}{\sqrt{\pi}} \cdot \left(\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)\right)\right|} \]
4. Step-by-step derivation

   1. div-inv 99.8%

      \[\leadsto \left|\color{blue}{\left(\left|x\right| \cdot \frac{1}{\sqrt{\pi}}\right)} \cdot \left(\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)\right)\right| \]

   2. pow1/2 99.8%

      \[\leadsto \left|\left(\left|x\right| \cdot \frac{1}{\color{blue}{{\pi}^{0.5}}}\right) \cdot \left(\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)\right)\right| \]

   3. pow-flip 99.8%

      \[\leadsto \left|\left(\left|x\right| \cdot \color{blue}{{\pi}^{\left(-0.5\right)}}\right) \cdot \left(\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)\right)\right| \]

   4. metadata-eval 99.8%

      \[\leadsto \left|\left(\left|x\right| \cdot {\pi}^{\color{blue}{-0.5}}\right) \cdot \left(\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)\right)\right| \]

5. Applied egg-rr 99.8%

   \[\leadsto \left|\color{blue}{\left(\left|x\right| \cdot {\pi}^{-0.5}\right)} \cdot \left(\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)\right)\right| \]
6. Step-by-step derivation

   1. unpow1 99.8%

      \[\leadsto \left|\left(\left|\color{blue}{{x}^{1}}\right| \cdot {\pi}^{-0.5}\right) \cdot \left(\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)\right)\right| \]

   2. sqr-pow 30.6%

      \[\leadsto \left|\left(\left|\color{blue}{{x}^{\left(\frac{1}{2}\right)} \cdot {x}^{\left(\frac{1}{2}\right)}}\right| \cdot {\pi}^{-0.5}\right) \cdot \left(\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)\right)\right| \]

   3. fabs-sqr 30.6%

      \[\leadsto \left|\left(\color{blue}{\left({x}^{\left(\frac{1}{2}\right)} \cdot {x}^{\left(\frac{1}{2}\right)}\right)} \cdot {\pi}^{-0.5}\right) \cdot \left(\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)\right)\right| \]

   4. sqr-pow 99.8%

      \[\leadsto \left|\left(\color{blue}{{x}^{1}} \cdot {\pi}^{-0.5}\right) \cdot \left(\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)\right)\right| \]

   5. unpow1 99.8%

      \[\leadsto \left|\left(\color{blue}{x} \cdot {\pi}^{-0.5}\right) \cdot \left(\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)\right)\right| \]

7. Simplified 99.8%

   \[\leadsto \left|\color{blue}{\left(x \cdot {\pi}^{-0.5}\right)} \cdot \left(\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)\right)\right| \]
8. Step-by-step derivation

   1. fma-udef 99.8%

      \[\leadsto \left|\left(x \cdot {\pi}^{-0.5}\right) \cdot \left(\color{blue}{\left(0.2 \cdot {x}^{4} + 0.047619047619047616 \cdot {x}^{6}\right)} + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)\right)\right| \]

9. Applied egg-rr 99.8%

   \[\leadsto \left|\left(x \cdot {\pi}^{-0.5}\right) \cdot \left(\color{blue}{\left(0.2 \cdot {x}^{4} + 0.047619047619047616 \cdot {x}^{6}\right)} + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)\right)\right| \]
10. Step-by-step derivation

    1. expm1-log1p-u 63.6%

       \[\leadsto \left|\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(x \cdot {\pi}^{-0.5}\right)\right)} \cdot \left(\left(0.2 \cdot {x}^{4} + 0.047619047619047616 \cdot {x}^{6}\right) + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)\right)\right| \]

    2. expm1-udef 5.4%

       \[\leadsto \left|\color{blue}{\left(e^{\mathsf{log1p}\left(x \cdot {\pi}^{-0.5}\right)} - 1\right)} \cdot \left(\left(0.2 \cdot {x}^{4} + 0.047619047619047616 \cdot {x}^{6}\right) + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)\right)\right| \]

    3. add-exp-log 2.3%

       \[\leadsto \left|\left(e^{\mathsf{log1p}\left(\color{blue}{e^{\log \left(x \cdot {\pi}^{-0.5}\right)}}\right)} - 1\right) \cdot \left(\left(0.2 \cdot {x}^{4} + 0.047619047619047616 \cdot {x}^{6}\right) + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)\right)\right| \]

    4. add-exp-log 5.4%

       \[\leadsto \left|\left(e^{\mathsf{log1p}\left(\color{blue}{x \cdot {\pi}^{-0.5}}\right)} - 1\right) \cdot \left(\left(0.2 \cdot {x}^{4} + 0.047619047619047616 \cdot {x}^{6}\right) + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)\right)\right| \]

    5. metadata-eval 5.4%

       \[\leadsto \left|\left(e^{\mathsf{log1p}\left(x \cdot {\pi}^{\color{blue}{\left(\frac{-1}{2}\right)}}\right)} - 1\right) \cdot \left(\left(0.2 \cdot {x}^{4} + 0.047619047619047616 \cdot {x}^{6}\right) + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)\right)\right| \]

    6. sqrt-pow1 5.4%

       \[\leadsto \left|\left(e^{\mathsf{log1p}\left(x \cdot \color{blue}{\sqrt{{\pi}^{-1}}}\right)} - 1\right) \cdot \left(\left(0.2 \cdot {x}^{4} + 0.047619047619047616 \cdot {x}^{6}\right) + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)\right)\right| \]

    7. inv-pow 5.4%

       \[\leadsto \left|\left(e^{\mathsf{log1p}\left(x \cdot \sqrt{\color{blue}{\frac{1}{\pi}}}\right)} - 1\right) \cdot \left(\left(0.2 \cdot {x}^{4} + 0.047619047619047616 \cdot {x}^{6}\right) + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)\right)\right| \]

    8. sqrt-div 5.4%

       \[\leadsto \left|\left(e^{\mathsf{log1p}\left(x \cdot \color{blue}{\frac{\sqrt{1}}{\sqrt{\pi}}}\right)} - 1\right) \cdot \left(\left(0.2 \cdot {x}^{4} + 0.047619047619047616 \cdot {x}^{6}\right) + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)\right)\right| \]

    9. metadata-eval 5.4%

       \[\leadsto \left|\left(e^{\mathsf{log1p}\left(x \cdot \frac{\color{blue}{1}}{\sqrt{\pi}}\right)} - 1\right) \cdot \left(\left(0.2 \cdot {x}^{4} + 0.047619047619047616 \cdot {x}^{6}\right) + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)\right)\right| \]

    10. un-div-inv 5.4%

        \[\leadsto \left|\left(e^{\mathsf{log1p}\left(\color{blue}{\frac{x}{\sqrt{\pi}}}\right)} - 1\right) \cdot \left(\left(0.2 \cdot {x}^{4} + 0.047619047619047616 \cdot {x}^{6}\right) + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)\right)\right| \]

11. Applied egg-rr 5.4%

    \[\leadsto \left|\color{blue}{\left(e^{\mathsf{log1p}\left(\frac{x}{\sqrt{\pi}}\right)} - 1\right)} \cdot \left(\left(0.2 \cdot {x}^{4} + 0.047619047619047616 \cdot {x}^{6}\right) + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)\right)\right| \]
12. Step-by-step derivation

    1. expm1-def 63.2%

       \[\leadsto \left|\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{x}{\sqrt{\pi}}\right)\right)} \cdot \left(\left(0.2 \cdot {x}^{4} + 0.047619047619047616 \cdot {x}^{6}\right) + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)\right)\right| \]

    2. expm1-log1p 99.5%

       \[\leadsto \left|\color{blue}{\frac{x}{\sqrt{\pi}}} \cdot \left(\left(0.2 \cdot {x}^{4} + 0.047619047619047616 \cdot {x}^{6}\right) + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)\right)\right| \]

13. Simplified 99.5%

    \[\leadsto \left|\color{blue}{\frac{x}{\sqrt{\pi}}} \cdot \left(\left(0.2 \cdot {x}^{4} + 0.047619047619047616 \cdot {x}^{6}\right) + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)\right)\right| \]

14. Final simplification 99.5%

    \[\leadsto \left|\left(\left(0.2 \cdot {x}^{4} + 0.047619047619047616 \cdot {x}^{6}\right) + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)\right) \cdot \frac{x}{\sqrt{\pi}}\right| \]

Alternative 3: 99.1% accurate, 3.8× speedup

Math

\[\begin{array}{l} \\ \left|\left(x \cdot {\pi}^{-0.5}\right) \cdot \left(\left(0.2 \cdot {x}^{4} + 0.047619047619047616 \cdot {x}^{6}\right) + 2\right)\right| \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (fabs
  (*
   (* x (pow PI -0.5))
   (+ (+ (* 0.2 (pow x 4.0)) (* 0.047619047619047616 (pow x 6.0))) 2.0))))

C

double code(double x) {
	return fabs(((x * pow(((double) M_PI), -0.5)) * (((0.2 * pow(x, 4.0)) + (0.047619047619047616 * pow(x, 6.0))) + 2.0)));
}

Java

public static double code(double x) {
	return Math.abs(((x * Math.pow(Math.PI, -0.5)) * (((0.2 * Math.pow(x, 4.0)) + (0.047619047619047616 * Math.pow(x, 6.0))) + 2.0)));
}

Python

def code(x):
	return math.fabs(((x * math.pow(math.pi, -0.5)) * (((0.2 * math.pow(x, 4.0)) + (0.047619047619047616 * math.pow(x, 6.0))) + 2.0)))

Julia

function code(x)
	return abs(Float64(Float64(x * (pi ^ -0.5)) * Float64(Float64(Float64(0.2 * (x ^ 4.0)) + Float64(0.047619047619047616 * (x ^ 6.0))) + 2.0)))
end

MATLAB

function tmp = code(x)
	tmp = abs(((x * (pi ^ -0.5)) * (((0.2 * (x ^ 4.0)) + (0.047619047619047616 * (x ^ 6.0))) + 2.0)));
end

Wolfram

code[x_] := N[Abs[N[(N[(x * N[Power[Pi, -0.5], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(0.2 * N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision] + N[(0.047619047619047616 * N[Power[x, 6.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 99.8%

   \[\left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \frac{2}{3} \cdot \left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{5} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{21} \cdot \left(\left(\left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right| \]

3. Simplified 99.5%

   \[\leadsto \color{blue}{\left|\frac{\left|x\right|}{\sqrt{\pi}} \cdot \left(\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)\right)\right|} \]
4. Step-by-step derivation

   1. div-inv 99.8%

      \[\leadsto \left|\color{blue}{\left(\left|x\right| \cdot \frac{1}{\sqrt{\pi}}\right)} \cdot \left(\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)\right)\right| \]

   2. pow1/2 99.8%

      \[\leadsto \left|\left(\left|x\right| \cdot \frac{1}{\color{blue}{{\pi}^{0.5}}}\right) \cdot \left(\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)\right)\right| \]

   3. pow-flip 99.8%

      \[\leadsto \left|\left(\left|x\right| \cdot \color{blue}{{\pi}^{\left(-0.5\right)}}\right) \cdot \left(\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)\right)\right| \]

   4. metadata-eval 99.8%

      \[\leadsto \left|\left(\left|x\right| \cdot {\pi}^{\color{blue}{-0.5}}\right) \cdot \left(\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)\right)\right| \]

5. Applied egg-rr 99.8%

   \[\leadsto \left|\color{blue}{\left(\left|x\right| \cdot {\pi}^{-0.5}\right)} \cdot \left(\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)\right)\right| \]
6. Step-by-step derivation

   1. unpow1 99.8%

      \[\leadsto \left|\left(\left|\color{blue}{{x}^{1}}\right| \cdot {\pi}^{-0.5}\right) \cdot \left(\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)\right)\right| \]

   2. sqr-pow 30.6%

      \[\leadsto \left|\left(\left|\color{blue}{{x}^{\left(\frac{1}{2}\right)} \cdot {x}^{\left(\frac{1}{2}\right)}}\right| \cdot {\pi}^{-0.5}\right) \cdot \left(\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)\right)\right| \]

   3. fabs-sqr 30.6%

      \[\leadsto \left|\left(\color{blue}{\left({x}^{\left(\frac{1}{2}\right)} \cdot {x}^{\left(\frac{1}{2}\right)}\right)} \cdot {\pi}^{-0.5}\right) \cdot \left(\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)\right)\right| \]

   4. sqr-pow 99.8%

      \[\leadsto \left|\left(\color{blue}{{x}^{1}} \cdot {\pi}^{-0.5}\right) \cdot \left(\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)\right)\right| \]

   5. unpow1 99.8%

      \[\leadsto \left|\left(\color{blue}{x} \cdot {\pi}^{-0.5}\right) \cdot \left(\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)\right)\right| \]

7. Simplified 99.8%

   \[\leadsto \left|\color{blue}{\left(x \cdot {\pi}^{-0.5}\right)} \cdot \left(\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)\right)\right| \]
8. Step-by-step derivation

   1. fma-udef 99.8%

      \[\leadsto \left|\left(x \cdot {\pi}^{-0.5}\right) \cdot \left(\color{blue}{\left(0.2 \cdot {x}^{4} + 0.047619047619047616 \cdot {x}^{6}\right)} + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)\right)\right| \]

9. Applied egg-rr 99.8%

   \[\leadsto \left|\left(x \cdot {\pi}^{-0.5}\right) \cdot \left(\color{blue}{\left(0.2 \cdot {x}^{4} + 0.047619047619047616 \cdot {x}^{6}\right)} + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)\right)\right| \]

10. Taylor expanded in x around 0 98.9%

    \[\leadsto \left|\left(x \cdot {\pi}^{-0.5}\right) \cdot \left(\left(0.2 \cdot {x}^{4} + 0.047619047619047616 \cdot {x}^{6}\right) + \color{blue}{2}\right)\right| \]

11. Final simplification 98.9%

    \[\leadsto \left|\left(x \cdot {\pi}^{-0.5}\right) \cdot \left(\left(0.2 \cdot {x}^{4} + 0.047619047619047616 \cdot {x}^{6}\right) + 2\right)\right| \]

Alternative 4: 89.3% accurate, 3.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.85:\\ \;\;\;\;\left|\sqrt{\frac{1}{\pi}} \cdot \left(x \cdot \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{0.2 \cdot {x}^{5} + 0.047619047619047616 \cdot {x}^{7}}{\sqrt{\pi}}\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 1.85)
   (fabs (* (sqrt (/ 1.0 PI)) (* x (fma 0.6666666666666666 (* x x) 2.0))))
   (fabs
    (/
     (+ (* 0.2 (pow x 5.0)) (* 0.047619047619047616 (pow x 7.0)))
     (sqrt PI)))))

C

double code(double x) {
	double tmp;
	if (x <= 1.85) {
		tmp = fabs((sqrt((1.0 / ((double) M_PI))) * (x * fma(0.6666666666666666, (x * x), 2.0))));
	} else {
		tmp = fabs((((0.2 * pow(x, 5.0)) + (0.047619047619047616 * pow(x, 7.0))) / sqrt(((double) M_PI))));
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= 1.85)
		tmp = abs(Float64(sqrt(Float64(1.0 / pi)) * Float64(x * fma(0.6666666666666666, Float64(x * x), 2.0))));
	else
		tmp = abs(Float64(Float64(Float64(0.2 * (x ^ 5.0)) + Float64(0.047619047619047616 * (x ^ 7.0))) / sqrt(pi)));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 1.85], N[Abs[N[(N[Sqrt[N[(1.0 / Pi), $MachinePrecision]], $MachinePrecision] * N[(x * N[(0.6666666666666666 * N[(x * x), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[(N[(0.2 * N[Power[x, 5.0], $MachinePrecision]), $MachinePrecision] + N[(0.047619047619047616 * N[Power[x, 7.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.8500000000000001

   1. Initial program 99.8%

      \[\left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \frac{2}{3} \cdot \left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{5} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{21} \cdot \left(\left(\left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right| \]

   3. Simplified 99.8%

      \[\leadsto \color{blue}{\left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\mathsf{fma}\left(2, \left|x\right|, 0.6666666666666666 \cdot \left(\left|x\right| \cdot \left(x \cdot x\right)\right)\right) + 0.2 \cdot \left(\left(\left|x\right| \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.047619047619047616 \cdot \left(\left(\left(\left|x\right| \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right)\right|} \]

   4. Taylor expanded in x around 0 87.2%

      \[\leadsto \left|\color{blue}{0.6666666666666666 \cdot \left(\left({x}^{2} \cdot \left|x\right|\right) \cdot \sqrt{\frac{1}{\pi}}\right) + 2 \cdot \left(\sqrt{\frac{1}{\pi}} \cdot \left|x\right|\right)}\right| \]
   5. Step-by-step derivation

      1. associate-*r* 87.2%

         \[\leadsto \left|\color{blue}{\left(0.6666666666666666 \cdot \left({x}^{2} \cdot \left|x\right|\right)\right) \cdot \sqrt{\frac{1}{\pi}}} + 2 \cdot \left(\sqrt{\frac{1}{\pi}} \cdot \left|x\right|\right)\right| \]

      2. *-commutative 87.2%

         \[\leadsto \left|\left(0.6666666666666666 \cdot \color{blue}{\left(\left|x\right| \cdot {x}^{2}\right)}\right) \cdot \sqrt{\frac{1}{\pi}} + 2 \cdot \left(\sqrt{\frac{1}{\pi}} \cdot \left|x\right|\right)\right| \]

      3. unpow2 87.2%

         \[\leadsto \left|\left(0.6666666666666666 \cdot \left(\left|x\right| \cdot \color{blue}{\left(x \cdot x\right)}\right)\right) \cdot \sqrt{\frac{1}{\pi}} + 2 \cdot \left(\sqrt{\frac{1}{\pi}} \cdot \left|x\right|\right)\right| \]

      4. sqr-abs 87.2%

         \[\leadsto \left|\left(0.6666666666666666 \cdot \left(\left|x\right| \cdot \color{blue}{\left(\left|x\right| \cdot \left|x\right|\right)}\right)\right) \cdot \sqrt{\frac{1}{\pi}} + 2 \cdot \left(\sqrt{\frac{1}{\pi}} \cdot \left|x\right|\right)\right| \]

      5. cube-mult 87.2%

         \[\leadsto \left|\left(0.6666666666666666 \cdot \color{blue}{{\left(\left|x\right|\right)}^{3}}\right) \cdot \sqrt{\frac{1}{\pi}} + 2 \cdot \left(\sqrt{\frac{1}{\pi}} \cdot \left|x\right|\right)\right| \]

      6. *-commutative 87.2%

         \[\leadsto \left|\left(0.6666666666666666 \cdot {\left(\left|x\right|\right)}^{3}\right) \cdot \sqrt{\frac{1}{\pi}} + 2 \cdot \color{blue}{\left(\left|x\right| \cdot \sqrt{\frac{1}{\pi}}\right)}\right| \]

      7. associate-*r* 87.2%

         \[\leadsto \left|\left(0.6666666666666666 \cdot {\left(\left|x\right|\right)}^{3}\right) \cdot \sqrt{\frac{1}{\pi}} + \color{blue}{\left(2 \cdot \left|x\right|\right) \cdot \sqrt{\frac{1}{\pi}}}\right| \]

      8. distribute-rgt-in 87.2%

         \[\leadsto \left|\color{blue}{\sqrt{\frac{1}{\pi}} \cdot \left(0.6666666666666666 \cdot {\left(\left|x\right|\right)}^{3} + 2 \cdot \left|x\right|\right)}\right| \]

      9. cube-mult 87.2%

         \[\leadsto \left|\sqrt{\frac{1}{\pi}} \cdot \left(0.6666666666666666 \cdot \color{blue}{\left(\left|x\right| \cdot \left(\left|x\right| \cdot \left|x\right|\right)\right)} + 2 \cdot \left|x\right|\right)\right| \]

      10. sqr-abs 87.2%

          \[\leadsto \left|\sqrt{\frac{1}{\pi}} \cdot \left(0.6666666666666666 \cdot \left(\left|x\right| \cdot \color{blue}{\left(x \cdot x\right)}\right) + 2 \cdot \left|x\right|\right)\right| \]

      11. unpow2 87.2%

          \[\leadsto \left|\sqrt{\frac{1}{\pi}} \cdot \left(0.6666666666666666 \cdot \left(\left|x\right| \cdot \color{blue}{{x}^{2}}\right) + 2 \cdot \left|x\right|\right)\right| \]

      12. *-commutative 87.2%

          \[\leadsto \left|\sqrt{\frac{1}{\pi}} \cdot \left(0.6666666666666666 \cdot \color{blue}{\left({x}^{2} \cdot \left|x\right|\right)} + 2 \cdot \left|x\right|\right)\right| \]

      13. associate-*r* 87.2%

          \[\leadsto \left|\sqrt{\frac{1}{\pi}} \cdot \left(\color{blue}{\left(0.6666666666666666 \cdot {x}^{2}\right) \cdot \left|x\right|} + 2 \cdot \left|x\right|\right)\right| \]

   6. Simplified 87.2%

      \[\leadsto \left|\color{blue}{\sqrt{\frac{1}{\pi}} \cdot \left(x \cdot \mathsf{fma}\left(0.6666666666666666, {x}^{2}, 2\right)\right)}\right| \]
   7. Step-by-step derivation

      1. unpow2 87.2%

         \[\leadsto \left|\sqrt{\frac{1}{\pi}} \cdot \left(x \cdot \mathsf{fma}\left(0.6666666666666666, \color{blue}{x \cdot x}, 2\right)\right)\right| \]

   8. Applied egg-rr 87.2%

      \[\leadsto \left|\sqrt{\frac{1}{\pi}} \cdot \left(x \cdot \mathsf{fma}\left(0.6666666666666666, \color{blue}{x \cdot x}, 2\right)\right)\right| \]

   if 1.8500000000000001 < x

   1. Initial program 99.8%

      \[\left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \frac{2}{3} \cdot \left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{5} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{21} \cdot \left(\left(\left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right| \]

   3. Simplified 99.8%

      \[\leadsto \color{blue}{\left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\mathsf{fma}\left(2, \left|x\right|, 0.6666666666666666 \cdot \left(\left|x\right| \cdot \left(x \cdot x\right)\right)\right) + 0.2 \cdot \left(\left(\left|x\right| \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.047619047619047616 \cdot \left(\left(\left(\left|x\right| \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right)\right|} \]

   4. Taylor expanded in x around inf 39.7%

      \[\leadsto \left|\color{blue}{0.047619047619047616 \cdot \left(\left({x}^{6} \cdot \left|x\right|\right) \cdot \sqrt{\frac{1}{\pi}}\right) + 0.2 \cdot \left(\left({x}^{4} \cdot \left|x\right|\right) \cdot \sqrt{\frac{1}{\pi}}\right)}\right| \]
   5. Step-by-step derivation

      1. +-commutative 39.7%

         \[\leadsto \left|\color{blue}{0.2 \cdot \left(\left({x}^{4} \cdot \left|x\right|\right) \cdot \sqrt{\frac{1}{\pi}}\right) + 0.047619047619047616 \cdot \left(\left({x}^{6} \cdot \left|x\right|\right) \cdot \sqrt{\frac{1}{\pi}}\right)}\right| \]

      2. associate-*r* 39.7%

         \[\leadsto \left|\color{blue}{\left(0.2 \cdot \left({x}^{4} \cdot \left|x\right|\right)\right) \cdot \sqrt{\frac{1}{\pi}}} + 0.047619047619047616 \cdot \left(\left({x}^{6} \cdot \left|x\right|\right) \cdot \sqrt{\frac{1}{\pi}}\right)\right| \]

      3. *-commutative 39.7%

         \[\leadsto \left|\left(0.2 \cdot \color{blue}{\left(\left|x\right| \cdot {x}^{4}\right)}\right) \cdot \sqrt{\frac{1}{\pi}} + 0.047619047619047616 \cdot \left(\left({x}^{6} \cdot \left|x\right|\right) \cdot \sqrt{\frac{1}{\pi}}\right)\right| \]

      4. associate-*r* 39.7%

         \[\leadsto \left|\left(0.2 \cdot \left(\left|x\right| \cdot {x}^{4}\right)\right) \cdot \sqrt{\frac{1}{\pi}} + \color{blue}{\left(0.047619047619047616 \cdot \left({x}^{6} \cdot \left|x\right|\right)\right) \cdot \sqrt{\frac{1}{\pi}}}\right| \]

      5. distribute-rgt-out 39.7%

         \[\leadsto \left|\color{blue}{\sqrt{\frac{1}{\pi}} \cdot \left(0.2 \cdot \left(\left|x\right| \cdot {x}^{4}\right) + 0.047619047619047616 \cdot \left({x}^{6} \cdot \left|x\right|\right)\right)}\right| \]

   6. Simplified 39.7%

      \[\leadsto \left|\color{blue}{\sqrt{\frac{1}{\pi}} \cdot \left(0.2 \cdot {x}^{5} + 0.047619047619047616 \cdot {x}^{7}\right)}\right| \]
   7. Step-by-step derivation

      1. expm1-log1p-u 3.7%

         \[\leadsto \left|\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\frac{1}{\pi}} \cdot \left(0.2 \cdot {x}^{5} + 0.047619047619047616 \cdot {x}^{7}\right)\right)\right)}\right| \]

      2. expm1-udef 3.3%

         \[\leadsto \left|\color{blue}{e^{\mathsf{log1p}\left(\sqrt{\frac{1}{\pi}} \cdot \left(0.2 \cdot {x}^{5} + 0.047619047619047616 \cdot {x}^{7}\right)\right)} - 1}\right| \]

      3. *-commutative 3.3%

         \[\leadsto \left|e^{\mathsf{log1p}\left(\color{blue}{\left(0.2 \cdot {x}^{5} + 0.047619047619047616 \cdot {x}^{7}\right) \cdot \sqrt{\frac{1}{\pi}}}\right)} - 1\right| \]

      4. sqrt-div 3.3%

         \[\leadsto \left|e^{\mathsf{log1p}\left(\left(0.2 \cdot {x}^{5} + 0.047619047619047616 \cdot {x}^{7}\right) \cdot \color{blue}{\frac{\sqrt{1}}{\sqrt{\pi}}}\right)} - 1\right| \]

      5. metadata-eval 3.3%

         \[\leadsto \left|e^{\mathsf{log1p}\left(\left(0.2 \cdot {x}^{5} + 0.047619047619047616 \cdot {x}^{7}\right) \cdot \frac{\color{blue}{1}}{\sqrt{\pi}}\right)} - 1\right| \]

      6. un-div-inv 3.3%

         \[\leadsto \left|e^{\mathsf{log1p}\left(\color{blue}{\frac{0.2 \cdot {x}^{5} + 0.047619047619047616 \cdot {x}^{7}}{\sqrt{\pi}}}\right)} - 1\right| \]

      7. +-commutative 3.3%

         \[\leadsto \left|e^{\mathsf{log1p}\left(\frac{\color{blue}{0.047619047619047616 \cdot {x}^{7} + 0.2 \cdot {x}^{5}}}{\sqrt{\pi}}\right)} - 1\right| \]

      8. fma-def 3.3%

         \[\leadsto \left|e^{\mathsf{log1p}\left(\frac{\color{blue}{\mathsf{fma}\left(0.047619047619047616, {x}^{7}, 0.2 \cdot {x}^{5}\right)}}{\sqrt{\pi}}\right)} - 1\right| \]

   8. Applied egg-rr 3.3%

      \[\leadsto \left|\color{blue}{e^{\mathsf{log1p}\left(\frac{\mathsf{fma}\left(0.047619047619047616, {x}^{7}, 0.2 \cdot {x}^{5}\right)}{\sqrt{\pi}}\right)} - 1}\right| \]
   9. Step-by-step derivation

      1. expm1-def 3.7%

         \[\leadsto \left|\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\mathsf{fma}\left(0.047619047619047616, {x}^{7}, 0.2 \cdot {x}^{5}\right)}{\sqrt{\pi}}\right)\right)}\right| \]

      2. expm1-log1p 39.8%

         \[\leadsto \left|\color{blue}{\frac{\mathsf{fma}\left(0.047619047619047616, {x}^{7}, 0.2 \cdot {x}^{5}\right)}{\sqrt{\pi}}}\right| \]

      3. fma-def 39.8%

         \[\leadsto \left|\frac{\color{blue}{0.047619047619047616 \cdot {x}^{7} + 0.2 \cdot {x}^{5}}}{\sqrt{\pi}}\right| \]

      4. +-commutative 39.8%

         \[\leadsto \left|\frac{\color{blue}{0.2 \cdot {x}^{5} + 0.047619047619047616 \cdot {x}^{7}}}{\sqrt{\pi}}\right| \]

      5. fma-udef 39.8%

         \[\leadsto \left|\frac{\color{blue}{\mathsf{fma}\left(0.2, {x}^{5}, 0.047619047619047616 \cdot {x}^{7}\right)}}{\sqrt{\pi}}\right| \]

   10. Simplified 39.8%

       \[\leadsto \left|\color{blue}{\frac{\mathsf{fma}\left(0.2, {x}^{5}, 0.047619047619047616 \cdot {x}^{7}\right)}{\sqrt{\pi}}}\right| \]
   11. Step-by-step derivation

       1. fma-udef 39.8%

          \[\leadsto \left|\frac{\color{blue}{0.2 \cdot {x}^{5} + 0.047619047619047616 \cdot {x}^{7}}}{\sqrt{\pi}}\right| \]

   12. Applied egg-rr 39.8%

       \[\leadsto \left|\frac{\color{blue}{0.2 \cdot {x}^{5} + 0.047619047619047616 \cdot {x}^{7}}}{\sqrt{\pi}}\right| \]
3. Recombined 2 regimes into one program.

4. Final simplification 87.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.85:\\ \;\;\;\;\left|\sqrt{\frac{1}{\pi}} \cdot \left(x \cdot \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{0.2 \cdot {x}^{5} + 0.047619047619047616 \cdot {x}^{7}}{\sqrt{\pi}}\right|\\ \end{array} \]

Alternative 5: 89.3% accurate, 4.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 2.2:\\ \;\;\;\;\left|\sqrt{\frac{1}{\pi}} \cdot \left(x \cdot \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|0.047619047619047616 \cdot \frac{{x}^{7}}{\sqrt{\pi}}\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 2.2)
   (fabs (* (sqrt (/ 1.0 PI)) (* x (fma 0.6666666666666666 (* x x) 2.0))))
   (fabs (* 0.047619047619047616 (/ (pow x 7.0) (sqrt PI))))))

C

double code(double x) {
	double tmp;
	if (x <= 2.2) {
		tmp = fabs((sqrt((1.0 / ((double) M_PI))) * (x * fma(0.6666666666666666, (x * x), 2.0))));
	} else {
		tmp = fabs((0.047619047619047616 * (pow(x, 7.0) / sqrt(((double) M_PI)))));
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= 2.2)
		tmp = abs(Float64(sqrt(Float64(1.0 / pi)) * Float64(x * fma(0.6666666666666666, Float64(x * x), 2.0))));
	else
		tmp = abs(Float64(0.047619047619047616 * Float64((x ^ 7.0) / sqrt(pi))));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 2.2], N[Abs[N[(N[Sqrt[N[(1.0 / Pi), $MachinePrecision]], $MachinePrecision] * N[(x * N[(0.6666666666666666 * N[(x * x), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(0.047619047619047616 * N[(N[Power[x, 7.0], $MachinePrecision] / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 2.2000000000000002

   1. Initial program 99.8%

      \[\left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \frac{2}{3} \cdot \left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{5} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{21} \cdot \left(\left(\left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right| \]

   3. Simplified 99.8%

      \[\leadsto \color{blue}{\left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\mathsf{fma}\left(2, \left|x\right|, 0.6666666666666666 \cdot \left(\left|x\right| \cdot \left(x \cdot x\right)\right)\right) + 0.2 \cdot \left(\left(\left|x\right| \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.047619047619047616 \cdot \left(\left(\left(\left|x\right| \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right)\right|} \]

   4. Taylor expanded in x around 0 87.2%

      \[\leadsto \left|\color{blue}{0.6666666666666666 \cdot \left(\left({x}^{2} \cdot \left|x\right|\right) \cdot \sqrt{\frac{1}{\pi}}\right) + 2 \cdot \left(\sqrt{\frac{1}{\pi}} \cdot \left|x\right|\right)}\right| \]
   5. Step-by-step derivation

      1. associate-*r* 87.2%

         \[\leadsto \left|\color{blue}{\left(0.6666666666666666 \cdot \left({x}^{2} \cdot \left|x\right|\right)\right) \cdot \sqrt{\frac{1}{\pi}}} + 2 \cdot \left(\sqrt{\frac{1}{\pi}} \cdot \left|x\right|\right)\right| \]

      2. *-commutative 87.2%

         \[\leadsto \left|\left(0.6666666666666666 \cdot \color{blue}{\left(\left|x\right| \cdot {x}^{2}\right)}\right) \cdot \sqrt{\frac{1}{\pi}} + 2 \cdot \left(\sqrt{\frac{1}{\pi}} \cdot \left|x\right|\right)\right| \]

      3. unpow2 87.2%

         \[\leadsto \left|\left(0.6666666666666666 \cdot \left(\left|x\right| \cdot \color{blue}{\left(x \cdot x\right)}\right)\right) \cdot \sqrt{\frac{1}{\pi}} + 2 \cdot \left(\sqrt{\frac{1}{\pi}} \cdot \left|x\right|\right)\right| \]

      4. sqr-abs 87.2%

         \[\leadsto \left|\left(0.6666666666666666 \cdot \left(\left|x\right| \cdot \color{blue}{\left(\left|x\right| \cdot \left|x\right|\right)}\right)\right) \cdot \sqrt{\frac{1}{\pi}} + 2 \cdot \left(\sqrt{\frac{1}{\pi}} \cdot \left|x\right|\right)\right| \]

      5. cube-mult 87.2%

         \[\leadsto \left|\left(0.6666666666666666 \cdot \color{blue}{{\left(\left|x\right|\right)}^{3}}\right) \cdot \sqrt{\frac{1}{\pi}} + 2 \cdot \left(\sqrt{\frac{1}{\pi}} \cdot \left|x\right|\right)\right| \]

      6. *-commutative 87.2%

         \[\leadsto \left|\left(0.6666666666666666 \cdot {\left(\left|x\right|\right)}^{3}\right) \cdot \sqrt{\frac{1}{\pi}} + 2 \cdot \color{blue}{\left(\left|x\right| \cdot \sqrt{\frac{1}{\pi}}\right)}\right| \]

      7. associate-*r* 87.2%

         \[\leadsto \left|\left(0.6666666666666666 \cdot {\left(\left|x\right|\right)}^{3}\right) \cdot \sqrt{\frac{1}{\pi}} + \color{blue}{\left(2 \cdot \left|x\right|\right) \cdot \sqrt{\frac{1}{\pi}}}\right| \]

      8. distribute-rgt-in 87.2%

         \[\leadsto \left|\color{blue}{\sqrt{\frac{1}{\pi}} \cdot \left(0.6666666666666666 \cdot {\left(\left|x\right|\right)}^{3} + 2 \cdot \left|x\right|\right)}\right| \]

      9. cube-mult 87.2%

         \[\leadsto \left|\sqrt{\frac{1}{\pi}} \cdot \left(0.6666666666666666 \cdot \color{blue}{\left(\left|x\right| \cdot \left(\left|x\right| \cdot \left|x\right|\right)\right)} + 2 \cdot \left|x\right|\right)\right| \]

      10. sqr-abs 87.2%

          \[\leadsto \left|\sqrt{\frac{1}{\pi}} \cdot \left(0.6666666666666666 \cdot \left(\left|x\right| \cdot \color{blue}{\left(x \cdot x\right)}\right) + 2 \cdot \left|x\right|\right)\right| \]

      11. unpow2 87.2%

          \[\leadsto \left|\sqrt{\frac{1}{\pi}} \cdot \left(0.6666666666666666 \cdot \left(\left|x\right| \cdot \color{blue}{{x}^{2}}\right) + 2 \cdot \left|x\right|\right)\right| \]

      12. *-commutative 87.2%

          \[\leadsto \left|\sqrt{\frac{1}{\pi}} \cdot \left(0.6666666666666666 \cdot \color{blue}{\left({x}^{2} \cdot \left|x\right|\right)} + 2 \cdot \left|x\right|\right)\right| \]

      13. associate-*r* 87.2%

          \[\leadsto \left|\sqrt{\frac{1}{\pi}} \cdot \left(\color{blue}{\left(0.6666666666666666 \cdot {x}^{2}\right) \cdot \left|x\right|} + 2 \cdot \left|x\right|\right)\right| \]

   6. Simplified 87.2%

      \[\leadsto \left|\color{blue}{\sqrt{\frac{1}{\pi}} \cdot \left(x \cdot \mathsf{fma}\left(0.6666666666666666, {x}^{2}, 2\right)\right)}\right| \]
   7. Step-by-step derivation

      1. unpow2 87.2%

         \[\leadsto \left|\sqrt{\frac{1}{\pi}} \cdot \left(x \cdot \mathsf{fma}\left(0.6666666666666666, \color{blue}{x \cdot x}, 2\right)\right)\right| \]

   8. Applied egg-rr 87.2%

      \[\leadsto \left|\sqrt{\frac{1}{\pi}} \cdot \left(x \cdot \mathsf{fma}\left(0.6666666666666666, \color{blue}{x \cdot x}, 2\right)\right)\right| \]

   if 2.2000000000000002 < x

   1. Initial program 99.8%

      \[\left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \frac{2}{3} \cdot \left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{5} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{21} \cdot \left(\left(\left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right| \]

   3. Simplified 99.8%

      \[\leadsto \color{blue}{\left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\mathsf{fma}\left(2, \left|x\right|, 0.6666666666666666 \cdot \left(\left|x\right| \cdot \left(x \cdot x\right)\right)\right) + 0.2 \cdot \left(\left(\left|x\right| \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.047619047619047616 \cdot \left(\left(\left(\left|x\right| \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right)\right|} \]

   4. Taylor expanded in x around inf 39.3%

      \[\leadsto \left|\color{blue}{0.047619047619047616 \cdot \left(\left({x}^{6} \cdot \left|x\right|\right) \cdot \sqrt{\frac{1}{\pi}}\right)}\right| \]
   5. Step-by-step derivation

      1. expm1-log1p-u 38.9%

         \[\leadsto \left|0.047619047619047616 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left({x}^{6} \cdot \left|x\right|\right) \cdot \sqrt{\frac{1}{\pi}}\right)\right)}\right| \]

      2. expm1-udef 38.6%

         \[\leadsto \left|0.047619047619047616 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\left({x}^{6} \cdot \left|x\right|\right) \cdot \sqrt{\frac{1}{\pi}}\right)} - 1\right)}\right| \]

      3. associate-*l* 38.6%

         \[\leadsto \left|0.047619047619047616 \cdot \left(e^{\mathsf{log1p}\left(\color{blue}{{x}^{6} \cdot \left(\left|x\right| \cdot \sqrt{\frac{1}{\pi}}\right)}\right)} - 1\right)\right| \]

      4. *-commutative 38.6%

         \[\leadsto \left|0.047619047619047616 \cdot \left(e^{\mathsf{log1p}\left(\color{blue}{\left(\left|x\right| \cdot \sqrt{\frac{1}{\pi}}\right) \cdot {x}^{6}}\right)} - 1\right)\right| \]

      5. add-sqr-sqrt 1.5%

         \[\leadsto \left|0.047619047619047616 \cdot \left(e^{\mathsf{log1p}\left(\left(\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right| \cdot \sqrt{\frac{1}{\pi}}\right) \cdot {x}^{6}\right)} - 1\right)\right| \]

      6. fabs-sqr 1.5%

         \[\leadsto \left|0.047619047619047616 \cdot \left(e^{\mathsf{log1p}\left(\left(\color{blue}{\left(\sqrt{x} \cdot \sqrt{x}\right)} \cdot \sqrt{\frac{1}{\pi}}\right) \cdot {x}^{6}\right)} - 1\right)\right| \]

      7. add-sqr-sqrt 3.3%

         \[\leadsto \left|0.047619047619047616 \cdot \left(e^{\mathsf{log1p}\left(\left(\color{blue}{x} \cdot \sqrt{\frac{1}{\pi}}\right) \cdot {x}^{6}\right)} - 1\right)\right| \]

      8. sqrt-div 3.3%

         \[\leadsto \left|0.047619047619047616 \cdot \left(e^{\mathsf{log1p}\left(\left(x \cdot \color{blue}{\frac{\sqrt{1}}{\sqrt{\pi}}}\right) \cdot {x}^{6}\right)} - 1\right)\right| \]

      9. metadata-eval 3.3%

         \[\leadsto \left|0.047619047619047616 \cdot \left(e^{\mathsf{log1p}\left(\left(x \cdot \frac{\color{blue}{1}}{\sqrt{\pi}}\right) \cdot {x}^{6}\right)} - 1\right)\right| \]

      10. un-div-inv 3.3%

          \[\leadsto \left|0.047619047619047616 \cdot \left(e^{\mathsf{log1p}\left(\color{blue}{\frac{x}{\sqrt{\pi}}} \cdot {x}^{6}\right)} - 1\right)\right| \]

   6. Applied egg-rr 3.3%

      \[\leadsto \left|0.047619047619047616 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{x}{\sqrt{\pi}} \cdot {x}^{6}\right)} - 1\right)}\right| \]
   7. Step-by-step derivation

      1. expm1-def 3.5%

         \[\leadsto \left|0.047619047619047616 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{x}{\sqrt{\pi}} \cdot {x}^{6}\right)\right)}\right| \]

      2. expm1-log1p 39.3%

         \[\leadsto \left|0.047619047619047616 \cdot \color{blue}{\left(\frac{x}{\sqrt{\pi}} \cdot {x}^{6}\right)}\right| \]

      3. *-commutative 39.3%

         \[\leadsto \left|0.047619047619047616 \cdot \color{blue}{\left({x}^{6} \cdot \frac{x}{\sqrt{\pi}}\right)}\right| \]

      4. associate-*r/ 39.3%

         \[\leadsto \left|0.047619047619047616 \cdot \color{blue}{\frac{{x}^{6} \cdot x}{\sqrt{\pi}}}\right| \]

      5. pow-plus 39.3%

         \[\leadsto \left|0.047619047619047616 \cdot \frac{\color{blue}{{x}^{\left(6 + 1\right)}}}{\sqrt{\pi}}\right| \]

      6. metadata-eval 39.3%

         \[\leadsto \left|0.047619047619047616 \cdot \frac{{x}^{\color{blue}{7}}}{\sqrt{\pi}}\right| \]

   8. Simplified 39.3%

      \[\leadsto \left|0.047619047619047616 \cdot \color{blue}{\frac{{x}^{7}}{\sqrt{\pi}}}\right| \]
3. Recombined 2 regimes into one program.

4. Final simplification 87.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2.2:\\ \;\;\;\;\left|\sqrt{\frac{1}{\pi}} \cdot \left(x \cdot \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|0.047619047619047616 \cdot \frac{{x}^{7}}{\sqrt{\pi}}\right|\\ \end{array} \]

Alternative 6: 67.9% accurate, 4.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.85:\\ \;\;\;\;\left|x \cdot \frac{2}{\sqrt{\pi}}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\sqrt{\frac{{x}^{14}}{\pi} \cdot 0.0022675736961451248}\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 1.85)
   (fabs (* x (/ 2.0 (sqrt PI))))
   (fabs (sqrt (* (/ (pow x 14.0) PI) 0.0022675736961451248)))))

C

double code(double x) {
	double tmp;
	if (x <= 1.85) {
		tmp = fabs((x * (2.0 / sqrt(((double) M_PI)))));
	} else {
		tmp = fabs(sqrt(((pow(x, 14.0) / ((double) M_PI)) * 0.0022675736961451248)));
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= 1.85) {
		tmp = Math.abs((x * (2.0 / Math.sqrt(Math.PI))));
	} else {
		tmp = Math.abs(Math.sqrt(((Math.pow(x, 14.0) / Math.PI) * 0.0022675736961451248)));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 1.85:
		tmp = math.fabs((x * (2.0 / math.sqrt(math.pi))))
	else:
		tmp = math.fabs(math.sqrt(((math.pow(x, 14.0) / math.pi) * 0.0022675736961451248)))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 1.85)
		tmp = abs(Float64(x * Float64(2.0 / sqrt(pi))));
	else
		tmp = abs(sqrt(Float64(Float64((x ^ 14.0) / pi) * 0.0022675736961451248)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 1.85)
		tmp = abs((x * (2.0 / sqrt(pi))));
	else
		tmp = abs(sqrt((((x ^ 14.0) / pi) * 0.0022675736961451248)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 1.85], N[Abs[N[(x * N[(2.0 / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[Sqrt[N[(N[(N[Power[x, 14.0], $MachinePrecision] / Pi), $MachinePrecision] * 0.0022675736961451248), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.8500000000000001

   1. Initial program 99.8%

      \[\left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \frac{2}{3} \cdot \left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{5} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{21} \cdot \left(\left(\left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right| \]

   3. Simplified 99.8%

      \[\leadsto \color{blue}{\left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\mathsf{fma}\left(2, \left|x\right|, 0.6666666666666666 \cdot \left(\left|x\right| \cdot \left(x \cdot x\right)\right)\right) + 0.2 \cdot \left(\left(\left|x\right| \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.047619047619047616 \cdot \left(\left(\left(\left|x\right| \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right)\right|} \]

   4. Taylor expanded in x around 0 64.9%

      \[\leadsto \left|\color{blue}{2 \cdot \left(\sqrt{\frac{1}{\pi}} \cdot \left|x\right|\right)}\right| \]
   5. Step-by-step derivation

      1. *-commutative 64.9%

         \[\leadsto \left|2 \cdot \color{blue}{\left(\left|x\right| \cdot \sqrt{\frac{1}{\pi}}\right)}\right| \]

      2. *-commutative 64.9%

         \[\leadsto \left|\color{blue}{\left(\left|x\right| \cdot \sqrt{\frac{1}{\pi}}\right) \cdot 2}\right| \]

      3. *-commutative 64.9%

         \[\leadsto \left|\color{blue}{\left(\sqrt{\frac{1}{\pi}} \cdot \left|x\right|\right)} \cdot 2\right| \]

      4. associate-*l* 64.9%

         \[\leadsto \left|\color{blue}{\sqrt{\frac{1}{\pi}} \cdot \left(\left|x\right| \cdot 2\right)}\right| \]

      5. unpow1 64.9%

         \[\leadsto \left|\sqrt{\frac{1}{\pi}} \cdot \left(\left|\color{blue}{{x}^{1}}\right| \cdot 2\right)\right| \]

      6. sqr-pow 30.5%

         \[\leadsto \left|\sqrt{\frac{1}{\pi}} \cdot \left(\left|\color{blue}{{x}^{\left(\frac{1}{2}\right)} \cdot {x}^{\left(\frac{1}{2}\right)}}\right| \cdot 2\right)\right| \]

      7. fabs-sqr 30.5%

         \[\leadsto \left|\sqrt{\frac{1}{\pi}} \cdot \left(\color{blue}{\left({x}^{\left(\frac{1}{2}\right)} \cdot {x}^{\left(\frac{1}{2}\right)}\right)} \cdot 2\right)\right| \]

      8. sqr-pow 64.9%

         \[\leadsto \left|\sqrt{\frac{1}{\pi}} \cdot \left(\color{blue}{{x}^{1}} \cdot 2\right)\right| \]

      9. unpow1 64.9%

         \[\leadsto \left|\sqrt{\frac{1}{\pi}} \cdot \left(\color{blue}{x} \cdot 2\right)\right| \]

   6. Simplified 64.9%

      \[\leadsto \left|\color{blue}{\sqrt{\frac{1}{\pi}} \cdot \left(x \cdot 2\right)}\right| \]
   7. Step-by-step derivation

      1. *-commutative 64.9%

         \[\leadsto \left|\color{blue}{\left(x \cdot 2\right) \cdot \sqrt{\frac{1}{\pi}}}\right| \]

      2. sqrt-div 64.9%

         \[\leadsto \left|\left(x \cdot 2\right) \cdot \color{blue}{\frac{\sqrt{1}}{\sqrt{\pi}}}\right| \]

      3. metadata-eval 64.9%

         \[\leadsto \left|\left(x \cdot 2\right) \cdot \frac{\color{blue}{1}}{\sqrt{\pi}}\right| \]

      4. un-div-inv 64.6%

         \[\leadsto \left|\color{blue}{\frac{x \cdot 2}{\sqrt{\pi}}}\right| \]

   8. Applied egg-rr 64.6%

      \[\leadsto \left|\color{blue}{\frac{x \cdot 2}{\sqrt{\pi}}}\right| \]
   9. Step-by-step derivation

      1. associate-*l/ 64.6%

         \[\leadsto \left|\color{blue}{\frac{x}{\sqrt{\pi}} \cdot 2}\right| \]

      2. expm1-log1p-u 62.5%

         \[\leadsto \left|\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{x}{\sqrt{\pi}} \cdot 2\right)\right)}\right| \]

      3. expm1-udef 5.1%

         \[\leadsto \left|\color{blue}{e^{\mathsf{log1p}\left(\frac{x}{\sqrt{\pi}} \cdot 2\right)} - 1}\right| \]

      4. associate-*l/ 5.1%

         \[\leadsto \left|e^{\mathsf{log1p}\left(\color{blue}{\frac{x \cdot 2}{\sqrt{\pi}}}\right)} - 1\right| \]

   10. Applied egg-rr 5.1%

       \[\leadsto \left|\color{blue}{e^{\mathsf{log1p}\left(\frac{x \cdot 2}{\sqrt{\pi}}\right)} - 1}\right| \]
   11. Step-by-step derivation

       1. expm1-def 62.5%

          \[\leadsto \left|\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{x \cdot 2}{\sqrt{\pi}}\right)\right)}\right| \]

       2. expm1-log1p 64.6%

          \[\leadsto \left|\color{blue}{\frac{x \cdot 2}{\sqrt{\pi}}}\right| \]

       3. *-commutative 64.6%

          \[\leadsto \left|\frac{\color{blue}{2 \cdot x}}{\sqrt{\pi}}\right| \]

       4. associate-/l* 64.5%

          \[\leadsto \left|\color{blue}{\frac{2}{\frac{\sqrt{\pi}}{x}}}\right| \]

   12. Simplified 64.5%

       \[\leadsto \left|\color{blue}{\frac{2}{\frac{\sqrt{\pi}}{x}}}\right| \]
   13. Step-by-step derivation

       1. associate-/r/ 64.9%

          \[\leadsto \left|\color{blue}{\frac{2}{\sqrt{\pi}} \cdot x}\right| \]

   14. Applied egg-rr 64.9%

       \[\leadsto \left|\color{blue}{\frac{2}{\sqrt{\pi}} \cdot x}\right| \]

   if 1.8500000000000001 < x

   1. Initial program 99.8%

      \[\left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \frac{2}{3} \cdot \left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{5} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{21} \cdot \left(\left(\left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right| \]

   3. Simplified 99.8%

      \[\leadsto \color{blue}{\left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\mathsf{fma}\left(2, \left|x\right|, 0.6666666666666666 \cdot \left(\left|x\right| \cdot \left(x \cdot x\right)\right)\right) + 0.2 \cdot \left(\left(\left|x\right| \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.047619047619047616 \cdot \left(\left(\left(\left|x\right| \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right)\right|} \]

   4. Taylor expanded in x around inf 39.7%

      \[\leadsto \left|\color{blue}{0.047619047619047616 \cdot \left(\left({x}^{6} \cdot \left|x\right|\right) \cdot \sqrt{\frac{1}{\pi}}\right) + 0.2 \cdot \left(\left({x}^{4} \cdot \left|x\right|\right) \cdot \sqrt{\frac{1}{\pi}}\right)}\right| \]
   5. Step-by-step derivation

      1. +-commutative 39.7%

         \[\leadsto \left|\color{blue}{0.2 \cdot \left(\left({x}^{4} \cdot \left|x\right|\right) \cdot \sqrt{\frac{1}{\pi}}\right) + 0.047619047619047616 \cdot \left(\left({x}^{6} \cdot \left|x\right|\right) \cdot \sqrt{\frac{1}{\pi}}\right)}\right| \]

      2. associate-*r* 39.7%

         \[\leadsto \left|\color{blue}{\left(0.2 \cdot \left({x}^{4} \cdot \left|x\right|\right)\right) \cdot \sqrt{\frac{1}{\pi}}} + 0.047619047619047616 \cdot \left(\left({x}^{6} \cdot \left|x\right|\right) \cdot \sqrt{\frac{1}{\pi}}\right)\right| \]

      3. *-commutative 39.7%

         \[\leadsto \left|\left(0.2 \cdot \color{blue}{\left(\left|x\right| \cdot {x}^{4}\right)}\right) \cdot \sqrt{\frac{1}{\pi}} + 0.047619047619047616 \cdot \left(\left({x}^{6} \cdot \left|x\right|\right) \cdot \sqrt{\frac{1}{\pi}}\right)\right| \]

      4. associate-*r* 39.7%

         \[\leadsto \left|\left(0.2 \cdot \left(\left|x\right| \cdot {x}^{4}\right)\right) \cdot \sqrt{\frac{1}{\pi}} + \color{blue}{\left(0.047619047619047616 \cdot \left({x}^{6} \cdot \left|x\right|\right)\right) \cdot \sqrt{\frac{1}{\pi}}}\right| \]

      5. distribute-rgt-out 39.7%

         \[\leadsto \left|\color{blue}{\sqrt{\frac{1}{\pi}} \cdot \left(0.2 \cdot \left(\left|x\right| \cdot {x}^{4}\right) + 0.047619047619047616 \cdot \left({x}^{6} \cdot \left|x\right|\right)\right)}\right| \]

   6. Simplified 39.7%

      \[\leadsto \left|\color{blue}{\sqrt{\frac{1}{\pi}} \cdot \left(0.2 \cdot {x}^{5} + 0.047619047619047616 \cdot {x}^{7}\right)}\right| \]
   7. Step-by-step derivation

      1. expm1-log1p-u 3.7%

         \[\leadsto \left|\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\frac{1}{\pi}} \cdot \left(0.2 \cdot {x}^{5} + 0.047619047619047616 \cdot {x}^{7}\right)\right)\right)}\right| \]

      2. expm1-udef 3.3%

         \[\leadsto \left|\color{blue}{e^{\mathsf{log1p}\left(\sqrt{\frac{1}{\pi}} \cdot \left(0.2 \cdot {x}^{5} + 0.047619047619047616 \cdot {x}^{7}\right)\right)} - 1}\right| \]

      3. *-commutative 3.3%

         \[\leadsto \left|e^{\mathsf{log1p}\left(\color{blue}{\left(0.2 \cdot {x}^{5} + 0.047619047619047616 \cdot {x}^{7}\right) \cdot \sqrt{\frac{1}{\pi}}}\right)} - 1\right| \]

      4. sqrt-div 3.3%

         \[\leadsto \left|e^{\mathsf{log1p}\left(\left(0.2 \cdot {x}^{5} + 0.047619047619047616 \cdot {x}^{7}\right) \cdot \color{blue}{\frac{\sqrt{1}}{\sqrt{\pi}}}\right)} - 1\right| \]

      5. metadata-eval 3.3%

         \[\leadsto \left|e^{\mathsf{log1p}\left(\left(0.2 \cdot {x}^{5} + 0.047619047619047616 \cdot {x}^{7}\right) \cdot \frac{\color{blue}{1}}{\sqrt{\pi}}\right)} - 1\right| \]

      6. un-div-inv 3.3%

         \[\leadsto \left|e^{\mathsf{log1p}\left(\color{blue}{\frac{0.2 \cdot {x}^{5} + 0.047619047619047616 \cdot {x}^{7}}{\sqrt{\pi}}}\right)} - 1\right| \]

      7. +-commutative 3.3%

         \[\leadsto \left|e^{\mathsf{log1p}\left(\frac{\color{blue}{0.047619047619047616 \cdot {x}^{7} + 0.2 \cdot {x}^{5}}}{\sqrt{\pi}}\right)} - 1\right| \]

      8. fma-def 3.3%

         \[\leadsto \left|e^{\mathsf{log1p}\left(\frac{\color{blue}{\mathsf{fma}\left(0.047619047619047616, {x}^{7}, 0.2 \cdot {x}^{5}\right)}}{\sqrt{\pi}}\right)} - 1\right| \]

   8. Applied egg-rr 3.3%

      \[\leadsto \left|\color{blue}{e^{\mathsf{log1p}\left(\frac{\mathsf{fma}\left(0.047619047619047616, {x}^{7}, 0.2 \cdot {x}^{5}\right)}{\sqrt{\pi}}\right)} - 1}\right| \]
   9. Step-by-step derivation

      1. expm1-def 3.7%

         \[\leadsto \left|\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\mathsf{fma}\left(0.047619047619047616, {x}^{7}, 0.2 \cdot {x}^{5}\right)}{\sqrt{\pi}}\right)\right)}\right| \]

      2. expm1-log1p 39.8%

         \[\leadsto \left|\color{blue}{\frac{\mathsf{fma}\left(0.047619047619047616, {x}^{7}, 0.2 \cdot {x}^{5}\right)}{\sqrt{\pi}}}\right| \]

      3. fma-def 39.8%

         \[\leadsto \left|\frac{\color{blue}{0.047619047619047616 \cdot {x}^{7} + 0.2 \cdot {x}^{5}}}{\sqrt{\pi}}\right| \]

      4. +-commutative 39.8%

         \[\leadsto \left|\frac{\color{blue}{0.2 \cdot {x}^{5} + 0.047619047619047616 \cdot {x}^{7}}}{\sqrt{\pi}}\right| \]

      5. fma-udef 39.8%

         \[\leadsto \left|\frac{\color{blue}{\mathsf{fma}\left(0.2, {x}^{5}, 0.047619047619047616 \cdot {x}^{7}\right)}}{\sqrt{\pi}}\right| \]

   10. Simplified 39.8%

       \[\leadsto \left|\color{blue}{\frac{\mathsf{fma}\left(0.2, {x}^{5}, 0.047619047619047616 \cdot {x}^{7}\right)}{\sqrt{\pi}}}\right| \]

   11. Taylor expanded in x around inf 39.3%

       \[\leadsto \left|\frac{\color{blue}{0.047619047619047616 \cdot {x}^{7}}}{\sqrt{\pi}}\right| \]
   12. Step-by-step derivation

       1. add-sqr-sqrt 3.3%

          \[\leadsto \left|\color{blue}{\sqrt{\frac{0.047619047619047616 \cdot {x}^{7}}{\sqrt{\pi}}} \cdot \sqrt{\frac{0.047619047619047616 \cdot {x}^{7}}{\sqrt{\pi}}}}\right| \]

       2. sqrt-unprod 36.4%

          \[\leadsto \left|\color{blue}{\sqrt{\frac{0.047619047619047616 \cdot {x}^{7}}{\sqrt{\pi}} \cdot \frac{0.047619047619047616 \cdot {x}^{7}}{\sqrt{\pi}}}}\right| \]

       3. frac-times 36.4%

          \[\leadsto \left|\sqrt{\color{blue}{\frac{\left(0.047619047619047616 \cdot {x}^{7}\right) \cdot \left(0.047619047619047616 \cdot {x}^{7}\right)}{\sqrt{\pi} \cdot \sqrt{\pi}}}}\right| \]

       4. *-commutative 36.4%

          \[\leadsto \left|\sqrt{\frac{\color{blue}{\left({x}^{7} \cdot 0.047619047619047616\right)} \cdot \left(0.047619047619047616 \cdot {x}^{7}\right)}{\sqrt{\pi} \cdot \sqrt{\pi}}}\right| \]

       5. *-commutative 36.4%

          \[\leadsto \left|\sqrt{\frac{\left({x}^{7} \cdot 0.047619047619047616\right) \cdot \color{blue}{\left({x}^{7} \cdot 0.047619047619047616\right)}}{\sqrt{\pi} \cdot \sqrt{\pi}}}\right| \]

       6. swap-sqr 36.4%

          \[\leadsto \left|\sqrt{\frac{\color{blue}{\left({x}^{7} \cdot {x}^{7}\right) \cdot \left(0.047619047619047616 \cdot 0.047619047619047616\right)}}{\sqrt{\pi} \cdot \sqrt{\pi}}}\right| \]

       7. pow-prod-up 36.4%

          \[\leadsto \left|\sqrt{\frac{\color{blue}{{x}^{\left(7 + 7\right)}} \cdot \left(0.047619047619047616 \cdot 0.047619047619047616\right)}{\sqrt{\pi} \cdot \sqrt{\pi}}}\right| \]

       8. metadata-eval 36.4%

          \[\leadsto \left|\sqrt{\frac{{x}^{\color{blue}{14}} \cdot \left(0.047619047619047616 \cdot 0.047619047619047616\right)}{\sqrt{\pi} \cdot \sqrt{\pi}}}\right| \]

       9. metadata-eval 36.4%

          \[\leadsto \left|\sqrt{\frac{{x}^{14} \cdot \color{blue}{0.0022675736961451248}}{\sqrt{\pi} \cdot \sqrt{\pi}}}\right| \]

       10. add-sqr-sqrt 36.4%

           \[\leadsto \left|\sqrt{\frac{{x}^{14} \cdot 0.0022675736961451248}{\color{blue}{\pi}}}\right| \]

   13. Applied egg-rr 36.4%

       \[\leadsto \left|\color{blue}{\sqrt{\frac{{x}^{14} \cdot 0.0022675736961451248}{\pi}}}\right| \]
   14. Step-by-step derivation

       1. associate-/l* 36.4%

          \[\leadsto \left|\sqrt{\color{blue}{\frac{{x}^{14}}{\frac{\pi}{0.0022675736961451248}}}}\right| \]

       2. associate-/r/ 36.4%

          \[\leadsto \left|\sqrt{\color{blue}{\frac{{x}^{14}}{\pi} \cdot 0.0022675736961451248}}\right| \]

   15. Simplified 36.4%

       \[\leadsto \left|\color{blue}{\sqrt{\frac{{x}^{14}}{\pi} \cdot 0.0022675736961451248}}\right| \]
3. Recombined 2 regimes into one program.

4. Final simplification 64.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.85:\\ \;\;\;\;\left|x \cdot \frac{2}{\sqrt{\pi}}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\sqrt{\frac{{x}^{14}}{\pi} \cdot 0.0022675736961451248}\right|\\ \end{array} \]

Alternative 7: 67.9% accurate, 4.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.85:\\ \;\;\;\;\left|x \cdot \frac{2}{\sqrt{\pi}}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|0.047619047619047616 \cdot \frac{{x}^{7}}{\sqrt{\pi}}\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 1.85)
   (fabs (* x (/ 2.0 (sqrt PI))))
   (fabs (* 0.047619047619047616 (/ (pow x 7.0) (sqrt PI))))))

C

double code(double x) {
	double tmp;
	if (x <= 1.85) {
		tmp = fabs((x * (2.0 / sqrt(((double) M_PI)))));
	} else {
		tmp = fabs((0.047619047619047616 * (pow(x, 7.0) / sqrt(((double) M_PI)))));
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= 1.85) {
		tmp = Math.abs((x * (2.0 / Math.sqrt(Math.PI))));
	} else {
		tmp = Math.abs((0.047619047619047616 * (Math.pow(x, 7.0) / Math.sqrt(Math.PI))));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 1.85:
		tmp = math.fabs((x * (2.0 / math.sqrt(math.pi))))
	else:
		tmp = math.fabs((0.047619047619047616 * (math.pow(x, 7.0) / math.sqrt(math.pi))))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 1.85)
		tmp = abs(Float64(x * Float64(2.0 / sqrt(pi))));
	else
		tmp = abs(Float64(0.047619047619047616 * Float64((x ^ 7.0) / sqrt(pi))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 1.85)
		tmp = abs((x * (2.0 / sqrt(pi))));
	else
		tmp = abs((0.047619047619047616 * ((x ^ 7.0) / sqrt(pi))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 1.85], N[Abs[N[(x * N[(2.0 / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(0.047619047619047616 * N[(N[Power[x, 7.0], $MachinePrecision] / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.8500000000000001

   1. Initial program 99.8%

      \[\left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \frac{2}{3} \cdot \left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{5} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{21} \cdot \left(\left(\left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right| \]

   3. Simplified 99.8%

      \[\leadsto \color{blue}{\left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\mathsf{fma}\left(2, \left|x\right|, 0.6666666666666666 \cdot \left(\left|x\right| \cdot \left(x \cdot x\right)\right)\right) + 0.2 \cdot \left(\left(\left|x\right| \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.047619047619047616 \cdot \left(\left(\left(\left|x\right| \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right)\right|} \]

   4. Taylor expanded in x around 0 64.9%

      \[\leadsto \left|\color{blue}{2 \cdot \left(\sqrt{\frac{1}{\pi}} \cdot \left|x\right|\right)}\right| \]
   5. Step-by-step derivation

      1. *-commutative 64.9%

         \[\leadsto \left|2 \cdot \color{blue}{\left(\left|x\right| \cdot \sqrt{\frac{1}{\pi}}\right)}\right| \]

      2. *-commutative 64.9%

         \[\leadsto \left|\color{blue}{\left(\left|x\right| \cdot \sqrt{\frac{1}{\pi}}\right) \cdot 2}\right| \]

      3. *-commutative 64.9%

         \[\leadsto \left|\color{blue}{\left(\sqrt{\frac{1}{\pi}} \cdot \left|x\right|\right)} \cdot 2\right| \]

      4. associate-*l* 64.9%

         \[\leadsto \left|\color{blue}{\sqrt{\frac{1}{\pi}} \cdot \left(\left|x\right| \cdot 2\right)}\right| \]

      5. unpow1 64.9%

         \[\leadsto \left|\sqrt{\frac{1}{\pi}} \cdot \left(\left|\color{blue}{{x}^{1}}\right| \cdot 2\right)\right| \]

      6. sqr-pow 30.5%

         \[\leadsto \left|\sqrt{\frac{1}{\pi}} \cdot \left(\left|\color{blue}{{x}^{\left(\frac{1}{2}\right)} \cdot {x}^{\left(\frac{1}{2}\right)}}\right| \cdot 2\right)\right| \]

      7. fabs-sqr 30.5%

         \[\leadsto \left|\sqrt{\frac{1}{\pi}} \cdot \left(\color{blue}{\left({x}^{\left(\frac{1}{2}\right)} \cdot {x}^{\left(\frac{1}{2}\right)}\right)} \cdot 2\right)\right| \]

      8. sqr-pow 64.9%

         \[\leadsto \left|\sqrt{\frac{1}{\pi}} \cdot \left(\color{blue}{{x}^{1}} \cdot 2\right)\right| \]

      9. unpow1 64.9%

         \[\leadsto \left|\sqrt{\frac{1}{\pi}} \cdot \left(\color{blue}{x} \cdot 2\right)\right| \]

   6. Simplified 64.9%

      \[\leadsto \left|\color{blue}{\sqrt{\frac{1}{\pi}} \cdot \left(x \cdot 2\right)}\right| \]
   7. Step-by-step derivation

      1. *-commutative 64.9%

         \[\leadsto \left|\color{blue}{\left(x \cdot 2\right) \cdot \sqrt{\frac{1}{\pi}}}\right| \]

      2. sqrt-div 64.9%

         \[\leadsto \left|\left(x \cdot 2\right) \cdot \color{blue}{\frac{\sqrt{1}}{\sqrt{\pi}}}\right| \]

      3. metadata-eval 64.9%

         \[\leadsto \left|\left(x \cdot 2\right) \cdot \frac{\color{blue}{1}}{\sqrt{\pi}}\right| \]

      4. un-div-inv 64.6%

         \[\leadsto \left|\color{blue}{\frac{x \cdot 2}{\sqrt{\pi}}}\right| \]

   8. Applied egg-rr 64.6%

      \[\leadsto \left|\color{blue}{\frac{x \cdot 2}{\sqrt{\pi}}}\right| \]
   9. Step-by-step derivation

      1. associate-*l/ 64.6%

         \[\leadsto \left|\color{blue}{\frac{x}{\sqrt{\pi}} \cdot 2}\right| \]

      2. expm1-log1p-u 62.5%

         \[\leadsto \left|\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{x}{\sqrt{\pi}} \cdot 2\right)\right)}\right| \]

      3. expm1-udef 5.1%

         \[\leadsto \left|\color{blue}{e^{\mathsf{log1p}\left(\frac{x}{\sqrt{\pi}} \cdot 2\right)} - 1}\right| \]

      4. associate-*l/ 5.1%

         \[\leadsto \left|e^{\mathsf{log1p}\left(\color{blue}{\frac{x \cdot 2}{\sqrt{\pi}}}\right)} - 1\right| \]

   10. Applied egg-rr 5.1%

       \[\leadsto \left|\color{blue}{e^{\mathsf{log1p}\left(\frac{x \cdot 2}{\sqrt{\pi}}\right)} - 1}\right| \]
   11. Step-by-step derivation

       1. expm1-def 62.5%

          \[\leadsto \left|\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{x \cdot 2}{\sqrt{\pi}}\right)\right)}\right| \]

       2. expm1-log1p 64.6%

          \[\leadsto \left|\color{blue}{\frac{x \cdot 2}{\sqrt{\pi}}}\right| \]

       3. *-commutative 64.6%

          \[\leadsto \left|\frac{\color{blue}{2 \cdot x}}{\sqrt{\pi}}\right| \]

       4. associate-/l* 64.5%

          \[\leadsto \left|\color{blue}{\frac{2}{\frac{\sqrt{\pi}}{x}}}\right| \]

   12. Simplified 64.5%

       \[\leadsto \left|\color{blue}{\frac{2}{\frac{\sqrt{\pi}}{x}}}\right| \]
   13. Step-by-step derivation

       1. associate-/r/ 64.9%

          \[\leadsto \left|\color{blue}{\frac{2}{\sqrt{\pi}} \cdot x}\right| \]

   14. Applied egg-rr 64.9%

       \[\leadsto \left|\color{blue}{\frac{2}{\sqrt{\pi}} \cdot x}\right| \]

   if 1.8500000000000001 < x

   1. Initial program 99.8%

      \[\left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \frac{2}{3} \cdot \left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{5} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{21} \cdot \left(\left(\left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right| \]

   3. Simplified 99.8%

      \[\leadsto \color{blue}{\left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\mathsf{fma}\left(2, \left|x\right|, 0.6666666666666666 \cdot \left(\left|x\right| \cdot \left(x \cdot x\right)\right)\right) + 0.2 \cdot \left(\left(\left|x\right| \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.047619047619047616 \cdot \left(\left(\left(\left|x\right| \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right)\right|} \]

   4. Taylor expanded in x around inf 39.3%

      \[\leadsto \left|\color{blue}{0.047619047619047616 \cdot \left(\left({x}^{6} \cdot \left|x\right|\right) \cdot \sqrt{\frac{1}{\pi}}\right)}\right| \]
   5. Step-by-step derivation

      1. expm1-log1p-u 38.9%

         \[\leadsto \left|0.047619047619047616 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left({x}^{6} \cdot \left|x\right|\right) \cdot \sqrt{\frac{1}{\pi}}\right)\right)}\right| \]

      2. expm1-udef 38.6%

         \[\leadsto \left|0.047619047619047616 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\left({x}^{6} \cdot \left|x\right|\right) \cdot \sqrt{\frac{1}{\pi}}\right)} - 1\right)}\right| \]

      3. associate-*l* 38.6%

         \[\leadsto \left|0.047619047619047616 \cdot \left(e^{\mathsf{log1p}\left(\color{blue}{{x}^{6} \cdot \left(\left|x\right| \cdot \sqrt{\frac{1}{\pi}}\right)}\right)} - 1\right)\right| \]

      4. *-commutative 38.6%

         \[\leadsto \left|0.047619047619047616 \cdot \left(e^{\mathsf{log1p}\left(\color{blue}{\left(\left|x\right| \cdot \sqrt{\frac{1}{\pi}}\right) \cdot {x}^{6}}\right)} - 1\right)\right| \]

      5. add-sqr-sqrt 1.5%

         \[\leadsto \left|0.047619047619047616 \cdot \left(e^{\mathsf{log1p}\left(\left(\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right| \cdot \sqrt{\frac{1}{\pi}}\right) \cdot {x}^{6}\right)} - 1\right)\right| \]

      6. fabs-sqr 1.5%

         \[\leadsto \left|0.047619047619047616 \cdot \left(e^{\mathsf{log1p}\left(\left(\color{blue}{\left(\sqrt{x} \cdot \sqrt{x}\right)} \cdot \sqrt{\frac{1}{\pi}}\right) \cdot {x}^{6}\right)} - 1\right)\right| \]

      7. add-sqr-sqrt 3.3%

         \[\leadsto \left|0.047619047619047616 \cdot \left(e^{\mathsf{log1p}\left(\left(\color{blue}{x} \cdot \sqrt{\frac{1}{\pi}}\right) \cdot {x}^{6}\right)} - 1\right)\right| \]

      8. sqrt-div 3.3%

         \[\leadsto \left|0.047619047619047616 \cdot \left(e^{\mathsf{log1p}\left(\left(x \cdot \color{blue}{\frac{\sqrt{1}}{\sqrt{\pi}}}\right) \cdot {x}^{6}\right)} - 1\right)\right| \]

      9. metadata-eval 3.3%

         \[\leadsto \left|0.047619047619047616 \cdot \left(e^{\mathsf{log1p}\left(\left(x \cdot \frac{\color{blue}{1}}{\sqrt{\pi}}\right) \cdot {x}^{6}\right)} - 1\right)\right| \]

      10. un-div-inv 3.3%

          \[\leadsto \left|0.047619047619047616 \cdot \left(e^{\mathsf{log1p}\left(\color{blue}{\frac{x}{\sqrt{\pi}}} \cdot {x}^{6}\right)} - 1\right)\right| \]

   6. Applied egg-rr 3.3%

      \[\leadsto \left|0.047619047619047616 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{x}{\sqrt{\pi}} \cdot {x}^{6}\right)} - 1\right)}\right| \]
   7. Step-by-step derivation

      1. expm1-def 3.5%

         \[\leadsto \left|0.047619047619047616 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{x}{\sqrt{\pi}} \cdot {x}^{6}\right)\right)}\right| \]

      2. expm1-log1p 39.3%

         \[\leadsto \left|0.047619047619047616 \cdot \color{blue}{\left(\frac{x}{\sqrt{\pi}} \cdot {x}^{6}\right)}\right| \]

      3. *-commutative 39.3%

         \[\leadsto \left|0.047619047619047616 \cdot \color{blue}{\left({x}^{6} \cdot \frac{x}{\sqrt{\pi}}\right)}\right| \]

      4. associate-*r/ 39.3%

         \[\leadsto \left|0.047619047619047616 \cdot \color{blue}{\frac{{x}^{6} \cdot x}{\sqrt{\pi}}}\right| \]

      5. pow-plus 39.3%

         \[\leadsto \left|0.047619047619047616 \cdot \frac{\color{blue}{{x}^{\left(6 + 1\right)}}}{\sqrt{\pi}}\right| \]

      6. metadata-eval 39.3%

         \[\leadsto \left|0.047619047619047616 \cdot \frac{{x}^{\color{blue}{7}}}{\sqrt{\pi}}\right| \]

   8. Simplified 39.3%

      \[\leadsto \left|0.047619047619047616 \cdot \color{blue}{\frac{{x}^{7}}{\sqrt{\pi}}}\right| \]
3. Recombined 2 regimes into one program.

4. Final simplification 64.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.85:\\ \;\;\;\;\left|x \cdot \frac{2}{\sqrt{\pi}}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|0.047619047619047616 \cdot \frac{{x}^{7}}{\sqrt{\pi}}\right|\\ \end{array} \]

Alternative 8: 67.9% accurate, 6.4× speedup

Math

\[\begin{array}{l} \\ \left|x \cdot \frac{2}{\sqrt{\pi}}\right| \end{array} \]

FPCore

(FPCore (x) :precision binary64 (fabs (* x (/ 2.0 (sqrt PI)))))

C

double code(double x) {
	return fabs((x * (2.0 / sqrt(((double) M_PI)))));
}

Java

public static double code(double x) {
	return Math.abs((x * (2.0 / Math.sqrt(Math.PI))));
}

Python

def code(x):
	return math.fabs((x * (2.0 / math.sqrt(math.pi))))

Julia

function code(x)
	return abs(Float64(x * Float64(2.0 / sqrt(pi))))
end

MATLAB

function tmp = code(x)
	tmp = abs((x * (2.0 / sqrt(pi))));
end

Wolfram

code[x_] := N[Abs[N[(x * N[(2.0 / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 99.8%

   \[\left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \frac{2}{3} \cdot \left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{5} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{21} \cdot \left(\left(\left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right| \]

3. Simplified 99.8%

   \[\leadsto \color{blue}{\left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\mathsf{fma}\left(2, \left|x\right|, 0.6666666666666666 \cdot \left(\left|x\right| \cdot \left(x \cdot x\right)\right)\right) + 0.2 \cdot \left(\left(\left|x\right| \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.047619047619047616 \cdot \left(\left(\left(\left|x\right| \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right)\right|} \]

4. Taylor expanded in x around 0 64.9%

   \[\leadsto \left|\color{blue}{2 \cdot \left(\sqrt{\frac{1}{\pi}} \cdot \left|x\right|\right)}\right| \]
5. Step-by-step derivation

   1. *-commutative 64.9%

      \[\leadsto \left|2 \cdot \color{blue}{\left(\left|x\right| \cdot \sqrt{\frac{1}{\pi}}\right)}\right| \]

   2. *-commutative 64.9%

      \[\leadsto \left|\color{blue}{\left(\left|x\right| \cdot \sqrt{\frac{1}{\pi}}\right) \cdot 2}\right| \]

   3. *-commutative 64.9%

      \[\leadsto \left|\color{blue}{\left(\sqrt{\frac{1}{\pi}} \cdot \left|x\right|\right)} \cdot 2\right| \]

   4. associate-*l* 64.9%

      \[\leadsto \left|\color{blue}{\sqrt{\frac{1}{\pi}} \cdot \left(\left|x\right| \cdot 2\right)}\right| \]

   5. unpow1 64.9%

      \[\leadsto \left|\sqrt{\frac{1}{\pi}} \cdot \left(\left|\color{blue}{{x}^{1}}\right| \cdot 2\right)\right| \]

   6. sqr-pow 30.5%

      \[\leadsto \left|\sqrt{\frac{1}{\pi}} \cdot \left(\left|\color{blue}{{x}^{\left(\frac{1}{2}\right)} \cdot {x}^{\left(\frac{1}{2}\right)}}\right| \cdot 2\right)\right| \]

   7. fabs-sqr 30.5%

      \[\leadsto \left|\sqrt{\frac{1}{\pi}} \cdot \left(\color{blue}{\left({x}^{\left(\frac{1}{2}\right)} \cdot {x}^{\left(\frac{1}{2}\right)}\right)} \cdot 2\right)\right| \]

   8. sqr-pow 64.9%

      \[\leadsto \left|\sqrt{\frac{1}{\pi}} \cdot \left(\color{blue}{{x}^{1}} \cdot 2\right)\right| \]

   9. unpow1 64.9%

      \[\leadsto \left|\sqrt{\frac{1}{\pi}} \cdot \left(\color{blue}{x} \cdot 2\right)\right| \]

6. Simplified 64.9%

   \[\leadsto \left|\color{blue}{\sqrt{\frac{1}{\pi}} \cdot \left(x \cdot 2\right)}\right| \]
7. Step-by-step derivation

   1. *-commutative 64.9%

      \[\leadsto \left|\color{blue}{\left(x \cdot 2\right) \cdot \sqrt{\frac{1}{\pi}}}\right| \]

   2. sqrt-div 64.9%

      \[\leadsto \left|\left(x \cdot 2\right) \cdot \color{blue}{\frac{\sqrt{1}}{\sqrt{\pi}}}\right| \]

   3. metadata-eval 64.9%

      \[\leadsto \left|\left(x \cdot 2\right) \cdot \frac{\color{blue}{1}}{\sqrt{\pi}}\right| \]

   4. un-div-inv 64.6%

      \[\leadsto \left|\color{blue}{\frac{x \cdot 2}{\sqrt{\pi}}}\right| \]

8. Applied egg-rr 64.6%

   \[\leadsto \left|\color{blue}{\frac{x \cdot 2}{\sqrt{\pi}}}\right| \]
9. Step-by-step derivation

   1. associate-*l/ 64.6%

      \[\leadsto \left|\color{blue}{\frac{x}{\sqrt{\pi}} \cdot 2}\right| \]

   2. expm1-log1p-u 62.5%

      \[\leadsto \left|\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{x}{\sqrt{\pi}} \cdot 2\right)\right)}\right| \]

   3. expm1-udef 5.1%

      \[\leadsto \left|\color{blue}{e^{\mathsf{log1p}\left(\frac{x}{\sqrt{\pi}} \cdot 2\right)} - 1}\right| \]

   4. associate-*l/ 5.1%

      \[\leadsto \left|e^{\mathsf{log1p}\left(\color{blue}{\frac{x \cdot 2}{\sqrt{\pi}}}\right)} - 1\right| \]

10. Applied egg-rr 5.1%

    \[\leadsto \left|\color{blue}{e^{\mathsf{log1p}\left(\frac{x \cdot 2}{\sqrt{\pi}}\right)} - 1}\right| \]
11. Step-by-step derivation

    1. expm1-def 62.5%

       \[\leadsto \left|\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{x \cdot 2}{\sqrt{\pi}}\right)\right)}\right| \]

    2. expm1-log1p 64.6%

       \[\leadsto \left|\color{blue}{\frac{x \cdot 2}{\sqrt{\pi}}}\right| \]

    3. *-commutative 64.6%

       \[\leadsto \left|\frac{\color{blue}{2 \cdot x}}{\sqrt{\pi}}\right| \]

    4. associate-/l* 64.5%

       \[\leadsto \left|\color{blue}{\frac{2}{\frac{\sqrt{\pi}}{x}}}\right| \]

12. Simplified 64.5%

    \[\leadsto \left|\color{blue}{\frac{2}{\frac{\sqrt{\pi}}{x}}}\right| \]
13. Step-by-step derivation

    1. associate-/r/ 64.9%

       \[\leadsto \left|\color{blue}{\frac{2}{\sqrt{\pi}} \cdot x}\right| \]

14. Applied egg-rr 64.9%

    \[\leadsto \left|\color{blue}{\frac{2}{\sqrt{\pi}} \cdot x}\right| \]

15. Final simplification 64.9%

    \[\leadsto \left|x \cdot \frac{2}{\sqrt{\pi}}\right| \]

Reproduce

herbie shell --seed 2023307 
(FPCore (x)
  :name "Jmat.Real.erfi, branch x less than or equal to 0.5"
  :precision binary64
  :pre (<= x 0.5)
  (fabs (* (/ 1.0 (sqrt PI)) (+ (+ (+ (* 2.0 (fabs x)) (* (/ 2.0 3.0) (* (* (fabs x) (fabs x)) (fabs x)))) (* (/ 1.0 5.0) (* (* (* (* (fabs x) (fabs x)) (fabs x)) (fabs x)) (fabs x)))) (* (/ 1.0 21.0) (* (* (* (* (* (* (fabs x) (fabs x)) (fabs x)) (fabs x)) (fabs x)) (fabs x)) (fabs x)))))))